Hi,
Insert will add new rows in your new table, it will not update the existing rows.
For this you will have to use update query as below
UPDATE T1 SET NOTE_MEMO_CARETECH=T2.NOTE_MEMO_CARETECH
FROM COPYCLAIMPRGNOTE1 T1
INNER JOIN COPYCLAIMPRGNOTE T2 ON T2.CLAIM_ID=T1.CLAIM_ID